Causal Characteristic Impedance of Planar Transmission Lines Dylan F

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IEEE TRANSACTIONS ON ADVANCED PACKAGING, VOL. 26, NO. 2, MAY 2003 165 Causal Characteristic Impedance of Planar Transmission Lines Dylan F. Williams, Fellow, IEEE, Bradley K. Alpert, Uwe Arz, Member, IEEE, David K. Walker, Member, IEEE, and Hartmut Grabinski

Abstract—We compute power-voltage, power-current, and quickly with the spectral-domain algorithm they employed to causal definitions of the characteristic impedance of microstrip calculate it. and coplanar-waveguide transmission lines on insulating and In 1978, Jansen [7] observed similar differences between conducting silicon substrates, and compare to measurement. characteristic impedances defined with different definitions. Index Terms—Causal, characteristic impedance, circuit theory, Jansen finally chose a voltage-current definition for reasons of coplanar waveguide, microstrip. “numerical efficiency.” In 1978, Bianco, et al. [8] performed a careful study of the issue, examining the dependence of I. INTRODUCTION power-voltage, power-current, and voltage-current definitions of characteristic impedance on the path choices, and obtained E COMPUTE the traditional power-voltage and power- very different frequency behaviors. current definitions of the characteristic impedance [1], W In 1979, Getsinger [9] argued that the characteristic [2] of planar transmission lines on insulating and conductive impedance of a microstrip should be set equal to the wave silicon substrates with the full-wave method of [3] and com- impedance of a simplified longitudinal-section electric (LSE) pare them to the causal minimum-phase definition proposed in model of the microstrip line. Bianco, et al. [10] criticized [4]. Where possible we compare computed values of the causal Getsinger’s conclusions, arguing that there was no sound impedance to measurement. In all cases we find good agree- basis for choosing Getsinger’s LSE-mode wave-impedance ment. The microstrip simulations have been reported in confer- definition over any other definition. ence [5]. In 1982, Jansen and Koster [11] argued that the definition Classical waveguide circuit theories define characteristic of characteristic impedance with the weakest frequency depen- impedance within the context of the circuit theory itself. That dence is best. On that basis, Jansen and Koster recommended is, they develop expressions for the voltage and current in terms using the power-current definition. of the fields in the guide, and then define the characteristic In 1991, Rautio [12] proposed a “three-dimensional defini- impedance of the guide as the ratio of that voltage and current tion” of characteristic impedance. This characteristic impedance when only the forward mode is present. This results in a unique is defined as that which best models the electrical behavior of a definition of characteristic impedance in terms of the wave microstrip line embedded in an idealized coaxial test fixture. Re- impedance. cently Zhu and Wu [13] attempted to refine Rautio’s approach. Classical waveguide circuit theories cannot be applied in What the early attempts at defining the characteristic planar transmission lines because they do not have a unique impedance of a microstrip transmission line lacked was a wave impedance. This has led to an animated debate in the suitable equivalent-circuit theory with well-defined properties literature over the relative merits of various definitions of to provide the context for the choice. The causal waveguide the characteristic impedance of microstrip and other planar circuit theory of [4], which avoids the TEM, TE, and TM transmission lines. restrictions of classical waveguide circuit theories, provides In 1975, Knorr and Tufekcioglu [6] observed that the just this context. power-current and power-voltage definitions of characteristic The causal waveguide circuit theory of [4] marries the power impedance did not agree well in microstrip lines they studied, normalization of [1] and [2] with additional constraints that en- fueling a debate over the appropriate choice of characteristic force simultaneity of the theory’s voltages and currents and the impedance in microstrip. They concluded that the power-cur- actual fields in the circuit. These additional constraints not only rent definition was preferable because it converged more guarantee that the network parameters of passive devices in this theory are causal, but a minimum-phase condition determines Manuscript received August 29, 2002; revised April 22, 2003. This work was the characteristic impedance of a single-mode waveguide supported in part by an ARFTG Student Fellowship. uniquely within a real positive frequency-independent multi- D. F. Williams, B. K. Alpert, and D. K. Walker are with the National plier. Institute of Standards and Technology, Boulder, CO 80303 USA (e-mail: [email protected]). This approach to defining characteristic impedance differs U. Arz was with the Universität Hannover, Hannover D-30167, Germany, and fundamentally from previous approaches. It replaces the often is now with the Physikalisch-Technische Bundesanstalt, Braunschweig, Ger- vague criteria employed in the past with a unique definition many. H. Grabinski is with the Universität Hannover, Hannover D-30167, Germany. based on the temporal properties and power normalization of Digital Object Identifier 10.1109/TADVP.2003.817339 the microwave circuit theory.

In [14] we examined some of the implications of the cir- where is the unit vector tangential to the integration path. cuit theory described in [4], determining the characteristic The phase angles of the characteristic impedances and impedance required by the minimum-phase constraint of that are equal to the phase angle of , which is a fixed property theory in a lossless coaxial waveguide, a lossless rectangular of the guide. This condition on the phase of the characteristic waveguide, and an infinitely wide metal-insulator-semicon- impedance is a consequence of the power-normalization of the ductor transmission line. In [5] we investigated microstrip circuit theory; it is required to ensure that the time-averaged lines of finite width on silicon substrates, using full-wave power in the guide is equal to the product of the voltage and calculations to compute the power-voltage and power-current the conjugate of the current [1]. definitions of the characteristic impedance of the microstrip The magnitude of the characteristic impedance is determined lines, and using a Hilbert-transform relationship to deter- by the choice of voltage or current path, and is therefore not mine the causal minimum-phase characteristic impedance. A defined uniquely by the traditional circuit theories of [1] and comparison showed that the minimum-phase characteristic [2]. impedance agrees well with some, but not all, of the conven- The causal circuit theory of [4] also imposes the power nor- tional definitions in microstrip lines. malization of [1], so the phase angle of the causal characteristic In this paper we expand on [5], treating microstrips and impedance is also equal to the phase angle of . However, coplanar waveguides on both lossless and lossy substrates, and the causal theory requires in addition that be minimum extending the study to include common measurement methods. phase, which implies that In each case studied, we compare the causal power-normalized definition to common conventional definitions considered (6) earlier by previous workers. We use these studies to show where is the Hilbert transform. This condition ensures that explicitly how the theory of [4] resolves the earlier debates cen- voltage (or current) excitations in the guide do not give rise to a tered around the best definition of the characteristic impedance current (or voltage) response before the excitation begins. of a planar transmission line. Finally, we demonstrate that the Once is determined by the power condition (3), the constant-capacitance method [15] and the calibration-compar- space of solutions for is defined by ison method [16] measure causal minimum-phase characteristic impedances. (7)

II. CHARACTERISTIC IMPEDANCE where is a real positive frequency-independent constant that determines the overall impedance normalization [4]. Equation Following [1] and [5] we define the power-voltage character- (7) results from two facts: the Hilbert transform has a null space istic impedance from consisting of the constant functions, and elsewhere the inverse of the Hilbert transform is its negative. (1) We fixed in (7) by matching and at a single frequency, as discussed in [4] and [17]. We used the lowest fre- and the power-current characteristic impedance from quency at which we had performed calculations to match to , because we believed our calculation errors to be smallest (2) there. where the complex power of the forward mode is III. COMPARISON OF DEFINITIONS We used the full-wave method of [3] to calculate the charac- (3) teristic impedance of the 5 wide microstrip line of Fig. 1. Fig. 2 compares this microstrip’s power-voltage, power-cur- is the angular frequency, is the unit vector in the direction rent, and causal minimum-phase definitions of characteristic of propagation, is the transverse coordinate, and impedance. are the transverse electric and magnetic fields of the forward The curve in Fig. 2 labeled “ ” is the magnitude of the char- mode, and the integral of Poynting’s vector in (3) is performed acteristic impedance determined from the phase of , which we over the entire cross section of the guide. The voltage of the calculated with the full-wave method of [3], and the minimum forward mode is found by integrating the electric field over a phase condition (6), as required by the causal circuit theory of path with [4]. The circles labeled “ (power/total-voltage)” are the mag- (4) nitudes of the characteristic impedance we calculated with the full-wave method [3] and defined with a power-voltage definition. Here the voltage integration path begins in the center of and current is found by integrating the magnetic filed over a the microstrip line at the ground plane on the back of the silicon closed path with substrate and terminates on the signal conductor on top of the oxide. This path corresponds to the solid vertical line in Fig. 1. (5) closed The squares labeled “ (power/center-conductor-current)” path correspond to the magnitudes of the characteristic impedance WILLIAMS et al.: CAUSAL CHARACTERISTIC IMPEDANCE OF PLANAR TRANSMISSION LINES 167

IV. RESOLVING PATH CHOICES

The traditional circuit theories of [1] and [2] do not uniquely specify the integration path used to define the voltage. This motivated the study by Bianco, et al. [8] of the effects of changing paths on the characteristic impedance of lossless microstrip lines. That study showed that path choice plays a significant role in determining the high-frequency behavior of the characteristic impedance. We will now demonstrate how the circuit theory of [4] resolves this issue of path choice in a microstrip line on a conductive silicon substrate. To illustrate how the theory of [4] can be used to resolve the debates centered around the definition of characteristic impedance, consider the voltage integration path beginning at the surface of the silicon substrate and going through the oxide Fig. 1. Microstrip line geometry. The signal conductor had a conductivity of 3 IH am, a width of 5 "m, and a thickness of 1 "m. The oxide layer was to the signal conductor. This path is just as consistent with the 1 "m thick and had a relative dielectric constant of 3.9. The silicon substrate was traditional circuit theories of [1] and [2] as the total-voltage "m 1 thick and had a relative dielectric constant of 11.7. The silicon substrate path we discussed earlier. Furthermore, one might imagine that, may be conductive. We call out the conductivity explicitly in the text and other figures. (From [5].) since devices in the transmission line are typically fabricated on the surface of the silicon substrate and connected directly between the surface of the silicon substrate and the signal conductor, that this voltage path corresponds most closely to the actual voltage across the device, and might be the best choice upon which to base the definition of characteristic impedance. Fig. 2 shows that the power/oxide-voltage characteristic impedance, which is labeled “ (Power/oxide-voltage)” and is defined with a voltage integration path corresponding only to that part of the total path in the oxide, agrees well with the power/total-voltage characteristic impedance at low frequencies, but differs significantly from the power/total-voltage definition at the high frequencies. This is because at low frequencies the electric-field lines terminate in charges at the surface of the silicon substrate, and so the voltage drop across the oxide equals the total voltage drop across the oxide and the substrate. At higher frequencies the surface charges cannot “follow” Fig. 2. Comparison of definitions of characteristic impedance of a microstrip line on a 100 m substrate. We matched j j with j j at 5 MHz by the electric field. That is, at higher frequencies, surface charges adjusting ! in (7). (From [17].). do not compensate the fields inside the substrate, and the electric field penetrates deeply into the silicon substrate. This is the “quasi-TEM” region of operation described by Hasegawa et al. we calculated with [3] using a current definition, where the in- [18]. As a result, the voltage drop across the oxide becomes tegration path used to determine the current exactly encloses the only a small fraction of the total voltage drop between the center microstrip center conductor on the top surface of the substrate. conductor and the ground, and a large discrepancy develops be- Although we did not plot the magnitude of the charac- tween the power/total-voltage and power/oxide-voltage characteristic impedance corresponding to the voltage-current defini- teristic impedances. tion on this or other plots, (1) and (2) imply The figure also shows that it is the power/total-voltage characteristic impedance that agrees most closely with the charac- (8) teristic impedance required by the minimum-phase condition and the causal circuit theory of [4]. It appears that the “partial” This shows that the magnitude of the voltage/current charac- voltage across the oxide does not produce a result consistent teristic impedance always lies between the magnitudes of the with the causal requirements of [4]: apparently not all voltage power/voltage and power/current impedances. Thus we see that paths are created equal! the causal, power/total-voltage, power/center-conductor-cur- To calculate the causal characteristic impedance from rent, and total-voltage/center-conductor-current definitions of the minimum-phase constraint of (6), we extrapolated our cal- characteristic impedance all agree closely in this microstrip. culated values of by assuming that they approach 0 This indicates that all of these conventional formulations are smoothly and uniformly at high frequencies, as described in consistent with the causal power-normalized characteristic [17]. Fig. 2 shows in dashed lines the bounds from [4] and [17] impedance required by the new circuit theory of [4]. on the total error we could have made in calculating due to 168 IEEE TRANSACTIONS ON ADVANCED PACKAGING, VOL. 26, NO. 2, MAY 2003

Fig. 4. Comparison of definitions of characteristic impedance of a 5 "m wide Fig. 3. Comparison of the inverse Fourier transforms of the causal and microstrip line on a substrate with a resistivity of 0.001 m. The dimensions power/oxide-voltage characteristic impedances. The power/oxide-voltage are shown in Fig. 1. definition predicts a response to current excitations before the excitation begins. unexpected behavior in at frequencies above 150 GHz. of the power/ground-plane-current and power/center-con- The fact that the magnitude of the power/oxide-voltage char- ductor-current characteristic impedances seen in Fig. 4 at high acteristic impedance falls well outside of these error bounds frequencies. confirms that the power/oxide-voltage definition cannot be min- Likewise, at the high frequencies, the return current in the sil- imum phase, and is thus not consistent with the causality con- icon closely matches the current carried by the center conductor. straints of [4]. However, at the low frequencies, where most of the return cur- Fig. 3 compares the inverse Fourier transforms of the causal rent flows in the ground plane, the return current in the substrate minimum-phase and power/oxide-voltage characteristic imped- becomes much smaller than the center-conductor current, and ances. The figure shows that the inverse Fourier transform of again we see large differences between definitions. is 0 for negative times, and starts at time 0, as expected. So, while the figure shows that the conventional The figure also shows that the inverse Fourier transform of power/total-voltage and power/center-conductor-current the power/oxide-voltage characteristic impedance begins well definitions studied before are close to , the power/sub- before time 0. Thus the power/oxide-voltage impedance defini- strate-current and the power/ground-plane-current definitions tion predicts that the voltage at the input of this microstrip line are not. Neither the “partial” return current in the ground will respond to a current excitation before the excitation begins. plane nor the partial return current in the substrate yield a This is clearly not physical, and will cause instabilities when characteristic impedance consistent with the requirements of this and other network parameters of associated circuit theory [4]. are used in conventional temporal simulations. While these examples may appear somewhat contrived, they Fig. 4 shows a similar study of power-current definitions do illustrate nicely how the causal power-normalized circuit on a highly conductive silicon substrate in which the substrate theory of [4] can be used to resolve the issues of path choice skin-effect plays an important role. The figure compares in planar transmission lines in a clear and unambiguous way. to the conventional power/total-voltage and power/center-con- Fig. 5 compares the causal, power/total-voltage, and ductor-current definitions, and to power/substrate-current power/center-conductor-current definitions for the microstrip and power/ground-plane-current definitions of characteristic line of Fig. 1 over a broad range of substrate conduc- impedance. In these two latter cases we set the current equal to tivity. Again, from relation (8) we see that the magnitude the return current solely in the silicon substrate or solely in the of the total-voltage/center-conductor-current character- perfect metal conductor on the back of the substrate. istic impedance will be just between the magnitude of the At low frequencies, the return current in the microstrip power/total-voltage, and power/center-conductor-current line takes the path of least resistance, which is in the per- definitions. The figure shows that all of these definitions of fect conductor on the back of the substrate. As a result, the characteristic impedance are in approximate agreement over power/ground-plane-current characteristic impedance agrees the entire frequency range. closely with the power/center-conductor-current characteristic However, the conventional power/total-voltage and impedance at low frequencies. power/center-conductor-current characteristic impedances The skin effect plays an important role in the current plotted in Fig. 5 are slightly different on low-loss substrates at distribution at high frequencies, and forces the return cur- high frequencies. Since [4] shows that the causal power-nor- rent to the surface of the silicon substrate as the frequency malized characteristic impedance is unique, and both of these increases. As a result, at high frequencies, the return current conventional definitions are power normalized, we conclude in the ground plane drops far below that of the total current that, to at least some extent, one or both of them violates carried by the center conductor. This results in the divergence causality. WILLIAMS et al.: CAUSAL CHARACTERISTIC IMPEDANCE OF PLANAR TRANSMISSION LINES 169

Fig. 6. Comparison of measured characteristic impedances of a coplanar waveguide and microstrip lines fabricated on a semi-insulating gallium-arsenide Fig. 5. Comparison of definitions of characteristic impedance of a 5 "m wide substrate with causal calculations. microstrip line on substrates of three different conductivities. We matched j j with j j at 1 GHz by adjusting ! in (7) to better illustrate the agreement between the definitions; the dimensions are shown in Fig. 1. (Lossy silicon data from [5].).

We determined in Fig. 5 from the Hilbert transform of its phase, which we had calculated with the full-wave method of [3] to 100 GHz. To perform the Hilbert transform we extrapolated the phase of smoothly to 0 at large frequencies. How- ever, errors in this phase extrapolation will result in errors in at lower frequencies. We calculated the error bounds given in [4] for errors in due to errors in this extrapolation, and found that the differences between the two conventional definitions were still somewhat below those bounds. From this we concluded that, while we can state with certainty that at least one of the two conventional definitions violates causality, we are, at least with our band-limited calculations, still unable to Fig. 7. Comparison of measured and calculated characteristic impedances for the silicon transmission lines described in [16]. distinguish which one does so. differed by more than 0.5 , and the differences were always V. M EASUREMENT below 0.05 above 250 MHz and less than to 0.02 above Where possible we measured the characteristic impedance 5 GHz. The measured phases never differed by more than 0.3 , and compared the result to . Fig. 6 compares calculated and were always below 0.05 above 100 MHz, and below 0.02 values of to the measured characteristic impedance of a above 5 GHz. We also performed the microstrip measurements coplanar waveguide (CPW) and a microstrip line fabricated on two different calibration sets on the same wafer, and found on semiinsulating gallium arsenide substrates. The CPW had a nearly constant magnitude offset of 0.7 . We also found a a71 wide center conductors separated from two 250 decreasing but systematic difference of the phases of the char- wide ground lines by 49 wide gaps. The gold conductors acteristic impedances on the two calibration sets. However, the were evaporated to a thickness of 0.5683 on a 500 differences in phase were less than 0.2 above 1 GHz, and less thick semi-insulating gallium-arsenide substrate, and had a than 0.03 above 10 GHz. We concluded from these compar- measured conductivity of 3.685 . The microstrip was isons that the random errors in the measurements were negli- fabricated on a 100 thick semi-insulating gallium-arsenide gible. substrate, and its center conductor was 75 wide, 0.964 We computed from calculated with the full-wave thick, and had a measured conductivity of 3.76 . method of [3]. The good agreement in the figure indicates that We used the measurement method of [15] to characterize the characteristic-impedance measurements are, indeed, causal the coplanar waveguide and microstrip from measurements of and power-normalized, and that the systematic errors in the their propagation constant and low-frequency capacitance. We measurements are small. measured the capacitance using a load and the method of [19], We also compared causal calculations to measurements of and measured the propagation constant with a multiline thru-re- the transmission lines fabricated on lossy silicon substrates de- flect-line calibration [20]. scribed in [16] and sketched in Fig. 7. These lines had a 50 To estimate our random measurement errors, we performed by 50 pad connected to a 10 wide center conductor the CPW and microstrip measurements twice. We found that fabricated on a 0.5 thick oxide layer grown on a silicon the measured magnitudes of the characteristic impedance never substrate with a resistivity of approximately 0.0125 . 170 IEEE TRANSACTIONS ON ADVANCED PACKAGING, VOL. 26, NO. 2, MAY 2003

The transmission line also employed two 20 wide metal a power-voltage definition of the characteristic impedance rails connected by a continuous 10 wide via through the consistent with the causal impedance required by the theory of oxide to a 10 wide ohmic contact to the silicon substrate. [4]. These coplanar-waveguide-like ground returns were fabricated at a distance of 100 from the microstrip-like center con- ACKNOWLEDGMENT ductor to reduce the resistance of the ground return through the The authors wish to thank J. Morgan for fabricating the substrate. However, even with these coplanar-waveguide-like coplanar waveguides and microstrip lines used in their experi- ground returns, our calculations show that the skin effect in the ments. substrate plays an important role in the transmission line behavior, increasing the resistance per unit length of the line from REFERENCES its dc value by approximately a factor of 5 at 25 GHz. These lines violate the constant-capacitance assumption em- [1] R. B. Marks and D. F. Williams, “A general waveguide circuit theory,” J. Res. Nat., Inst. Stand. Technol., vol. 97, no. 5, pp. 533–562, Sept.-Oct. ployed by the measurement method of [15], so we measured the 1992. characteristic impedance of the lines with the calibration-com- [2] N. Fache, F. Olyslager, and D. de Zutter, Electromagnetic and Circuit parison method described in [16]. This method is optimized to Modeling of Multiconductor Transmission Lines. Oxford, U.K.: Clarendon Press, 1993. account for the large capacitive parasitics typical of contact pads [3] W. Heinrich, “Full-wave analysis of conductor losses on MMIC trans- fabricated on silicon substrates [16]. mission lines,” IEEE Trans. Microwave Theory Tech., vol. MTT-38, pp. Fig. 7 compares these measurements to calculations per- 1468–1472, Oct. 1990. [4] D. F. Williams and B. K. Alpert, “Causality and waveguide circuit formed with the quasianalytic method of [21], and to a causal theory,” IEEE Trans. Microwave Theory Tech., vol. MTT-48, pp. impedance we calculated from the phase of determined 615–623, Apr. 2000. by [21]. The magnitudes are so close that they cannot be [5] , “Characteristic impedance of microstrip on silicon,” Proc. 8th Elect. Perf. Electron. Packag. Conf. Dig., pp. 181–184, Oct. 1999. distinguished on the graph. [6] J. B. Knorr and A. Tufekcioglu, “Spectral-domain calculation of We were not able to verify directly that the phase of microstrip characteristic impedance,” IEEE Trans. Microwave Theory calculated by the method of [21] was set equal to ,so Tech., vol. MTT-23, pp. 725–728, Sept. 1975. [7] R. H. Jansen, “High-speed computation of single and coupled microstrip we cannot say with certainty that the measured characteristic parameters including dispersion, high-order modes, loss and finite strip impedance is power-normalized. Nevertheless, the agreement thickness,” IEEE Trans. Microwave Theory Tech., vol. MTT-26, pp. between the causal magnitude calculation, the calculations of 75–82, Feb. 1978. [8] B. Bianco, L. Panini, M. Parodi, and S. Ridella, “Some considerations [21], and the measurements is close. While we were not able to about the frequency dependence of the characteristic impedance of uni- determine independently how large our random measurement form microstrips,” IEEE Trans. Microwave Theory Tech., vol. MTT-26, errors were, the results do indicate that both the measurement pp. 182–185, Mar. 1978. [9] W. J. Getsinger, “Microstrip characteristic impedance,” IEEE Trans. Mi- method of [16] and the calculation method of [21] determined crowave Theory Tech., vol. MTT-27, p. 293, Apr. 1979. causal minimum-phase characteristic impedances. We found [10] B. Bianco, M. Parodi, and S. Ridella, “Comments on ’Microstrip similar agreement between calculation and measurement for characteristic impedance’,” IEEE Trans. Microwave Theory Tech., vol. MTT-28, p. 152, Feb. 1980. lines fabricated on this substrate with center conductors 2- , [11] R. H. Jansen and H. L. Koster, “New aspects concerning the definition 5- , and 50- wide. of microstrip characteristic impedance as a function of frequency,” 1982 IEEE MTT Symp. Dig., pp. 305–307, June 1982. [12] J. C. Rautio, “A new definition of characteristic impedance,” 1991 IEEE MTT Symp. Dig., pp. 761–764, June 1982. VI. CONCLUSION [13] L. Zhu and K. Wu, “Revisiting characteristic impedance and its definition of microstrip line with a self-calibrations 3-D MoM scheme,” IEEE The causal power-normalized waveguide circuit theory of [4] Microwave Guided Wave Lett., vol. 8, pp. 87–89, Feb. 1998. ensures simultaneity of its voltages, currents, and fields. This [14] D. F. Williams and B. K. Alpert, “Characteristic impedance, power, and ensures that the network parameters retain a number of physical causality,” IEEE Microwave Guided Wave Lett., vol. 9, pp. 181–182, May 1999. temporal properties required for time-domain simulation. In [15] R. B. Marks and D. F. Williams, “Characteristic impedance determi- this work we have shown how this new theory resolves the nation using propagation constant measurement,” IEEE Microwave debate around the definition of characteristic impedance of Guided Wave Lett., vol. 1, pp. 141–143, June 1991. [16] D. F. Williams, U. Arz, and H. Grabinski, “Accurate characteristic planar transmission lines, showing that the minimum-phase impedance measurement on silicon,” 1998 IEEE MTT-S Symp. Dig., constraints on the characteristic impedance required by the pp. 1917–1920, June 1998. causal theory are met by some, but not all, of the power/voltage [17] D. F. Williams, R. C. Wittmann, and D. Sterling, “Computation of causal characteristic impedances,” 2000 IEEE MTT-S Int. Microwave Symp. and power/current definitions in the planar transmission lines Dig., pp. 1813–1816, June 2000. we studied. [18] H. Hasegawa, M. Furukawa, and H. Yanai, “Properties of microstrip In the cases studied, we also found that the measurement line on Si-iy system,” IEEE Trans. Microwave Theory Tech., vol. MTT-19, pp. 869–881, Nov. 1971. methods of [15] and [16] determined characteristic impedances [19] D. F. Williams and R. B. Marks, “Transmission line capacitance mea- consistent with the causal requirements of the circuit theory of surement,” IEEE Microwave Guided Wave Lett., vol. 1, pp. 243–245, [4]. Sept. 1991. [20] R. B. Marks, “A multiline method of network analyzer calibration,” We hope that more detailed comparisons of the causal IEEE Trans. Microwave Theory Tech., vol. 39, pp. 1205–1215, July power-normalized characteristic impedance of planar guides 1991. with conventional definitions will resolve the more subtle [21] E. Grotelüschen, L. S. Dutta, and S. Zaage, “Quasianalytical analysis of the broadband properties of multiconductor transmission lines on questions, such as exact starting and ending points in lossy semiconducting substrates,” IEEE Trans. Comp., Packag., Manufact. microstrip conductors for the voltage path required to render Technol. B, vol. 17, pp. 376–382, Aug. 1994. WILLIAMS et al.: CAUSAL CHARACTERISTIC IMPEDANCE OF PLANAR TRANSMISSION LINES 171

Dylan F. Williams (F’02) received the Ph.D. in electrical engineering from the David K. Walker (M’98) received the B.S. degree in physics and the B.S. de- University of California, Berkeley, in 1986. gree in mathematics from Hastings College, Hastings, NE, in 1980, and the He joined the Electromagnetic Fields Division, National Institute of Stan- B.S. and M.S. degrees in electrical engineering from Washington University, dards and Technology, Boulder, CO, in 1989 where he develops metrology for St. Louis, MO, in 1982 and ’83, respectively. the characterization of monolithic microwave integrated circuits and electronic He spent eight years in industry working on microwave semiconductor de- interconnects. He has published over 80 technical papers. vice design and fabrication before joining the National Institute of Standards Dr. Williams received the Department of Commerce Bronze and Silver and Technology (NIST), Boulder, CO, in 1991 as part of the MMIC Project Medals, the Electrical Engineering Laboratory’s Outstanding Paper Award, in the Microwave Metrology Group. His work at NIST has included semicon- two ARFTG Best Paper Awards, the ARFTG Automated Measurements ductor fabrication, network analyzer calibration, and on-wafer measurements. Technology Award, and the IEEE Morris E. Leeds Award. He is an Associate Current research interests include microwave thermal noise measurements and Editor of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. calibration of radiometers for remote sensing. He holds five patents related to microwave technology.

Bradley K. Alpert received the B.S. degree in mathematics from the University of Illinois, Urbana-Champaign, in 1978, the S.M. degree in mathematics from the University of Chicago, Chicago, IL, in 1979, and the Ph.D. degree in com- puter science from Yale University, Storrs, CT, in 1990. He was a Casualty Actuary for several years. He was the Hans Lewy Post- doctoral Fellow at Lawrence Berkeley Laboratory and University of California, Berkeley. In 1991, he joined the National Institute of Standards and Technology, Boulder, CO. He was Visiting Researcher at Courant Institute of Mathematical Sciences, New York University, from 1999 to 2000. His research interests are in scientific computing, including high-order convergent quadratures, the fast multipole method, wavelets, special functions, and computational electromag- netics. Dr. Alpert received the Bronze Medal from the Department of Commerce for work on antenna probe position correction, in 1997. Hartmut Grabinski was born in Hildesheim, Germany. He received the Dipl.-Ing. degree from the Fachhochschule Hannover, Hannover, Germany, in Uwe Arz (S’02–M’03) received the Dipl.-Ing. degree in electrical engineering 1977 and the Dipl.-Ing. and Dr.-Ing. degrees from the University of Hannover, and the Ph.D. degree (with highest honors) from the University of Hannover, Hannover, Germany, in 1982 and 1987, respectively. Hannover, Germany, in 1994 and 2001, respectively. In 1993, he concluded his professorial dissertation, and since that time he lec- From 1995 to 2001, he was a Research and Teaching Assistant at the Labo- tures in the field of electrodynamics at the University of Hannover. Presently, ratory of Information Technology, University of Hannover. In 2001, he served he is the Manager of the Division of Design and Test, Laboratory for Informa- as a Postdoctoral Research Associate at the National Institute of Standards and tion Technology, University of Hannover. In addition, he is holding lectures on Technology, Boulder, CO. In 2002, he joined the Physikalisch-Technische Bun- “Electrical Performance of Electronic Packaging” and “Relativistic Electrody- desanstalt (PTB), Braunschweig, Germany, where he develops metrology for namics.” His research interests are also in the field of electronic packaging and on-wafer measurements. electrodynamics. Dr. Arz received the first ARFTG Microwave Measurement Fellowship Dr. Grabinski is the Founder of the IEEE Workshop “Signal Propagation on Award. Interconnects.”